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Mirrors > Home > MPE Home > Th. List > ditgex | Structured version Visualization version GIF version |
Description: A directed integral is a set. (Contributed by Mario Carneiro, 7-Sep-2014.) |
Ref | Expression |
---|---|
ditgex | ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ditg 24447 | . 2 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) | |
2 | itgex 24373 | . . 3 ⊢ ∫(𝐴(,)𝐵)𝐶 d𝑥 ∈ V | |
3 | negex 10886 | . . 3 ⊢ -∫(𝐵(,)𝐴)𝐶 d𝑥 ∈ V | |
4 | 2, 3 | ifex 4517 | . 2 ⊢ if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) ∈ V |
5 | 1, 4 | eqeltri 2911 | 1 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 Vcvv 3496 ifcif 4469 class class class wbr 5068 (class class class)co 7158 ≤ cle 10678 -cneg 10873 (,)cioo 12741 ∫citg 24221 ⨜cdit 24446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-uni 4841 df-iota 6316 df-fv 6365 df-ov 7161 df-neg 10875 df-sum 15045 df-itg 24226 df-ditg 24447 |
This theorem is referenced by: itgsubstlem 24647 |
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